Additive properties of dense subsets of sifted sequences
type: Item
instance of: scholarly article
title: Additive properties of dense subsets of sifted sequences
zbMATH Open document ID: 0996.11057
author: Olivier Ramaré
author: Imre Z. Ruzsa
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2002-07-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93827
full work available at URL: jourelec
review text: The authors begin by defining a sufficiently sifted sequence \({\mathcal A}\) of integers with dimension \(\kappa\). \({\mathcal A}^*\subset{\mathcal A}\) is called a dense subsequence of \({\mathcal A}\) if \(A^*(X)\geq A(X)/k\) for \(X\geq X_1\) and fixed \(k\geq 1\), where \(A(X)\) denotes the counting function of \({\mathcal A}\). This paper is concerned with establishing in Corollary 1 that, under certain conditions, \({\mathcal A}^*\) is an additive asymptotic basis of the integers and obtaining an upper bound of its order of the shape \(O_{\mathcal A}(k(\log\log 3k)^\kappa)\). For \({\mathcal A}\) the sequence of primes, when \(\kappa=1\), the order of the corresponding basis \({\mathcal P}^*\) obtained by \textit{A. Sárközy} [Astérisque 258, 109-127 (1999; Zbl 0969.11003)] was \(O(k^4)\). The above results follow from the authors' main theorem in which an asymmetrical lower bound for \(({\mathcal A}_1^*+{\mathcal A}_2^*)(X)\) is found, where, for \(i=1,2\), \({\mathcal A}_i\) is sufficiently sifted with dimension \(\kappa_i\), \({\mathcal A}_i^*\subset {\mathcal A}_i\) and \(A_i^*(X)\geq A_i(X)/k_i\) for \(X\geq X_1\) and fixed \(k_i\geq 1\). The authors go on to consider a finite version of these problems for which additional notation is required; we omit the details. The proofs, which are very technical although classed as elementary, depend on a version of the Selberg sieve and of the large sieve discussed in section 4, and then the required consequences are derived.
Mathematics Subject Classification ID: 11N36
Mathematics Subject Classification ID: 11P99
Mathematics Subject Classification ID: 11B05
Mathematics Subject Classification ID: 11B13
zbMATH DE Number: 1770786
zbMATH Keywords: sufficiently sifted sequence
zbMATH Keywords: additive asymptotic basis
zbMATH Keywords: upper bound
zbMATH Keywords: asymmetrical lower bound
zbMATH Keywords: Selberg sieve
zbMATH Keywords: large sieve
reviewed by: Eira J. Scourfield
MaRDI profile type: Publication
OpenAlex ID: W2170121928
cites work: THE 'LARGE SIEVE' METHOD AND ITS APPLICATIONS IN THE THEORY OF NUMBERS
cites work: Q3754686
cites work: Q5607318
cites work: The Addition of Primes and Power
cites work: The large sieve
cites work: Sieving by prime powers
cites work: On the representation of a number in the form x^2+y^2+p^2+q^2 where p, q are odd primes
cites work: Q4052222
cites work: Q3963072
cites work: Scharfe untere Abschätzung für die Anzahlfunktion der B-Zwillinge
cites work: Primes of the type φ(x, y)+A where φ is a quadratic form
cites work: On primes in arithmetic progressions
cites work: Topics in multiplicative number theory
cites work: Lectures on sieve methods and prime number theory
cites work: Q4875069
cites work: On an additive property of squares and primes
cites work: Essential Components
cites work: Q4256474
cites work: Q4143412
DOI: 10.5802/JTNB.338
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description: scientific article; zbMATH DE number 1770786